Dynamic Eigen Decomposition III: Applications to Flutter Prediction, Linear Time Varying Systems

If it hasn't been clear to you what we mean by 'Modally Equivalent', it means that the solution spaces of the two systems can be spanned by the same set of modes that are not necessarily fixed in space but changing with time, i.e., the dynamic eigenmodes. This is quite a departure from the traditional definition of 'vector spaces' and 'spanning' but one can still derive the equivalence based on the moving basis vectors.

In the second half of the video, the whole point of the proof is that you can use as many frequency samples as you want making the approximation of the convolution integral as accurate as possible and in the limit you'll get the MEPS for the linear time-varying system.

It is important to know that the perturbed flutter equation has the rank of 2N, exactly twice the number of structural degrees of freedom, regardless of how many degrees of freedom the original aeroelastic equation of motion has. This means there are only 2N nonzero dynamic eigenvalues and we need to deal with only 2N dynamic eigenmodes. Typically, the coupled AE equation has far more fluid DOFs than the structural DOFs. For example, if we use a CFD (computational fluid dynamics) model the number of fluid equations can easily reach an order of 10^5 to 10^6. On the other hand, the structural equations are in a much smaller order, say of 10^2. Hence, by working with the perturbed equation we only need to calculate the 2N dynamic eigenmodes and eigenvalues instead of solving the 10^6+2N equations.

The biggest takeaway from the application of DED to flutter prediction is that flutter can be thought of as an intrinsic property of the aeroelastic system, rather than a solution of the system at the particular condition. Most interestingly, the flutter mode exists in the aeroelastic system and therefore can be extracted at any dynamic pressure values.